The first recorded location problem that I know of was approached
by the Emperor Constantine in the 4th century. Constantine's
challenge was to deploy the legions of Rome in such a way as to
secure and protect his empire from invasion or insurrection.

In the 3rd century, when Rome dominated Europe, it was able to
deploy 50 legions throughout the empire, securing even the
furthermost areas. By the following century the empire had lost
much of its muscle, however, and Rome's forces had diminished to
just 25 legions. Emperor Constantine's problem: How to station
legions in sufficient strength to protect the most forward
positions of the empire without abandoning the core--namely Rome.
He devised a new defensive strategy to cope with Rome's reduced
power.

The problem is not "solved" in a mathematical sense, but a set of
rules exists that defines when a solution is acceptable. Once
you understand the rules, you can attempt to see if you can
improve on Constantine's choice of deployment.

The Rules
Each set of six legions may be thought of as a "pebble," a unit
of forces whose presence is sufficient to secure any one of the
regions of the empire. The regions of the empire may be
considered to be connected as shown below. Moving along the line
between regions represents a "step," and for a region to be
securable, a pebble must be able to reach it in just one step--to
repel invaders or put down a revolt.

A region, then, may be thought of as either secured or securable.
It is considered to be secured if it has one or more pebbles
placed in it already. It is considered securable if a pebble can
be deployed to that region in a single step. At any shift or
movement from a region, two pebbles must initially be present
together before one of them can be launched. That is, a
pebble can only be deployed if it moves from an adjacent region
where there is already another pebble to help launch it. This is
analogous to the island hopping strategy pursued by General
MacArthur in World War II in the Pacific theater--where movement
only followed the chain of islands (secured areas).

Now that you know the rules, the challenge is to place just four
pebbles in the eight regions of the empire.

Constantine's strategy
Constantine's placement could not protect the entire empire. He
had to leave one region uncovered, that is unable to be reached,
according to the rules, in a single step. Constantine placed two
pebbles at Rome, a symbolic as well as strategic choice, and two
at his new capital, Constantinople. With this deployment, each
region of the empire could be reached by a pebble in just one
step--except for Britain. To reach Britain required a pebble to
move from Rome to Gaul, securing Gaul, and a second pebble to
move from Constantinople to Rome, then from Rome to Gaul, and
finally from Gaul to Britain, a total of four steps. It is no
wonder that Britain was lost.

Here is another alternative, not necessarily better than
Constantine's strategy, but it gives you an idea of
possibilities. We will place one pebble in Gaul, two in Rome,
and one in Constantinople. Britain can now be reached in two
steps (a pebble from Rome to Gaul and a pebble from Gaul to
Britain), better for Britain than before. However, Asia Minor is
now not reachable in one step, but two (from Rome to
Constantinople and Constantinople to Asia Minor). All the rest
of the empire is reachable in just one step. It is not clear
that this is better than Constantine's strategy. Although the
number of steps to the worst-off nodes has been reduced to two,
the number of regions more than one step away has gone from one
to two.

Can you improve on Constantine's solution?
If you would like to try, here's how to evaluate the merit of the
alternative. There are, for our purposes, two criteria. The
first is the number of regions that cannot be reached in a single
step. For Constantine's solution, that number is just one. The
second is the number of steps it takes to reach the worst-off
node. Again, for Constantine's choice, this number is four
steps--to reach Britain.

If you can keep the number of nodes that can't be reached in one
step to just one, and can reduce the maximum number of steps to
reach that node to a number less than four, then you have done
better than Constantine. Of course, you hit the jackpot if you
can make all regions either initially secure or reachable in one
step, given the rules.

I will tell you that it is possible to do better than
Constantine, but I won't tell you how. If you do have success at
allocating the pebbles, you should think about the consequences
of a second war occurring somewhere in the empire. Of the
situations you create, which would be better in the event of a
second war at one of the unsecured regions? The answer, which I
mailed in early February, is in a sealed envelope in the desk
drawer of editor Sue De Pasquale. She has agreed to publish it in
the next issue of the Johns Hopkins Magazine.

A final footnote
Although we can examine specific arrangements of a known number
of regions, to my knowledge, no universal mathematical procedure
has yet been published to solve this problem in the general case.
The general case would have any number of regions in any
arrangement and a number of pebbles specified in advance. I have
recently made progress on the problem with my colleague Ken
Rosing, progress we hope will be published soon. I would also
like to acknowledge the marvelous paper that introduced me to
this problem. It is "Graphing an Optimal Grand Strategy," by John
Arquilla and Hal Fredricksen, which appeared in the Fall 1995
Military Operations Research.